Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Introducing Various Notions of Distances between Space-Times

Anna Sakovich, Christina Sormani

TL;DR

The paper develops a comprehensive, time-aware framework for comparing space-times by canonically converting each space-time into a compact timed-metric-space via the cosmological time $\tau_g$ and the null distance $\hat{d}_g$. It introduces causally-null-compactifiable space-times and a family of intrinsic distances (timeless, level-based, big bang, future-developed, strip-based, and timed-Hausdorff) to study convergence to potentially non-smooth limits, proving definiteness for the timed-Hausdorff distance and several big bang/future-developed variants. It also constructs exhaustions by causally-null-compactifiable sub-spaces and extends the theory to timed metric measure and integral current space-times, outlining open questions and conjectures. The work provides a robust, time-respecting toolkit for stability and convergence questions in General Relativity, accommodating black holes and asymptotically flat regions while enabling precise comparisons across non-diffeomorphic space-times.

Abstract

We introduce the notion of causally-null-compactifiable space-times which can be canonically converted into a compact timed-metric-spaces using the cosmological time of Andersson-Howard-Galloway and the null distance of Sormani-Vega. We produce a large class of such space-times including future developments of compact initial data sets and regions which exhaust asymptotically flat space-times. We then present various notions of intrinsic distances between these space-times (introducing the timed-Hausdorff distance) and prove some of these notions of distance are definite in the sense that they equal zero iff there is a time-oriented Lorentzian isometry between the space-times. These definite distances enable us to define various notions of convergence of space-times to limit space-times which are not necessarily smooth. Many open questions and conjectures are included throughout.

Introducing Various Notions of Distances between Space-Times

TL;DR

The paper develops a comprehensive, time-aware framework for comparing space-times by canonically converting each space-time into a compact timed-metric-space via the cosmological time and the null distance . It introduces causally-null-compactifiable space-times and a family of intrinsic distances (timeless, level-based, big bang, future-developed, strip-based, and timed-Hausdorff) to study convergence to potentially non-smooth limits, proving definiteness for the timed-Hausdorff distance and several big bang/future-developed variants. It also constructs exhaustions by causally-null-compactifiable sub-spaces and extends the theory to timed metric measure and integral current space-times, outlining open questions and conjectures. The work provides a robust, time-respecting toolkit for stability and convergence questions in General Relativity, accommodating black holes and asymptotically flat regions while enabling precise comparisons across non-diffeomorphic space-times.

Abstract

We introduce the notion of causally-null-compactifiable space-times which can be canonically converted into a compact timed-metric-spaces using the cosmological time of Andersson-Howard-Galloway and the null distance of Sormani-Vega. We produce a large class of such space-times including future developments of compact initial data sets and regions which exhaust asymptotically flat space-times. We then present various notions of intrinsic distances between these space-times (introducing the timed-Hausdorff distance) and prove some of these notions of distance are definite in the sense that they equal zero iff there is a time-oriented Lorentzian isometry between the space-times. These definite distances enable us to define various notions of convergence of space-times to limit space-times which are not necessarily smooth. Many open questions and conjectures are included throughout.

Paper Structure

This paper contains 47 sections, 23 theorems, 355 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Background
  3. The Gromov-Hausdorff Distance between Riemannian Manifolds and Metric Spaces
  4. Review of Metric Completions
  5. Pointed Gromov-Hausdorff Convergence
  6. Review of Measures and Metric Measure Distances
  7. Integral Currents and the Intrinsic Flat Distance
  8. Review of Fréchet Maps
  9. Review of Lorentzian Manifolds and Cosmological Time
  10. Review of the Null Distance
  11. Review of Causality and Lorentzian Isometries
  12. Causally-Null-Compactifiable Space-Times
  13. Metric-Space-Times defined by Cosmic Strips
  14. Space-Times with Proper Regular Cosmological Time Functions
  15. Big Bang Space-Times
  16. ...and 32 more sections

Key Result

Theorem 2.2

The Gromov-Hausdorff distance between compact Riemannian manifolds is definite in the sense that implies that there is a distance preserving bijection, which is thus is also a Riemannian isometry,

Figures (12)

  • Figure 3.1: Past sets of points in $N$ of Example \ref{['ex:induce']} are easily computed. The past sets of antipodal points intersect allowing for easy estimation of $\hat{d}_{g}$.
  • Figure 3.2: Past sets of antipodal points in $N_{s,t}$ of Example \ref{['ex:induce']} do not intersect so we need to take many zig-zags to estimate $\hat{d}_{g,s,t}$ and see that this distance is only achieved by a curve that leaves $N_{s,t}$, and in fact, $\hat{d}_{g,s,t}>\hat{d}_{g}$.
  • Figure 3.3: Example \ref{['ex:proper']}: On the left we see the level sets of $\tau_g$ in $N$ and its generators in $N$ achieving the distance to the big bang point at the center. On the right, we see the generators in $N_{s,t}$ achieving the distance to the initial set on the inner ring, and sequences of generators converging to a generator starting at a point in $\bar{N}_{s,t}$ on the outer ring.
  • Figure 3.4: Here we see $q, q_1,q_2 \in N_{\tau_{min},\tau_{max},p}\subset {\mathbb R}^{1,3}$ from the proof of Proposition \ref{['prop:past-in-Mink']}.
  • Figure 3.5: Here we see $N_{0,\tau_{max},R}\subset {\mathbb R}^{1,3}$ of Example \ref{['ex:Mink-ext']} and a generator, $\gamma_q$, of its cosmological time.
  • ...and 7 more figures

Theorems & Definitions (186)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 2.1
  • Theorem 2.2: Gromov
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.6: Fréchet
  • proof
  • ...and 176 more